# Documentation

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# beta

Beta function

B = beta(Z,W)

## Definitions

The beta function is

$B\left(z,w\right)={\int }_{0}^{1}{t}^{z-1}{\left(1-t\right)}^{w-1}dt=\frac{\Gamma \left(z\right)\Gamma \left(w\right)}{\Gamma \left(z+w\right)}$

where Γ(z) is the gamma function.

## Description

B = beta(Z,W) computes the beta function for corresponding elements of arrays Z and W. The arrays must be real and nonnegative. They must be the same size, or either can be scalar.

## Examples

In this example, which uses integer arguments,

beta(n,3)
= (n-1)!*2!/(n+2)!
= 2/(n*(n+1)*(n+2))

is the ratio of fairly small integers, and the rational format is able to recover the exact result.

format rat
beta((0:10)',3)

ans =

1/0
1/3
1/12
1/30
1/60
1/105
1/168
1/252
1/360
1/495
1/660

collapse all

### Algorithms

beta(z,w) = exp(gammaln(z)+gammaln(w)-gammaln(z+w))